Introductory mathematics
:This article is a continuation of Elementary mathematics Believe it or not the basis of all of mathematics is nothing more than the simple function. :Next(0)=1 :Next(1)=2 :Next(2)=3 :Next(3)=4 This defines the . Natural numbers are those used for counting. :These have the very convenient property of being . That means that if a1-3=x for which there is no answer among natural numbers. To provide an answer mathematicians generalize to the set of all which includes negative integers. :The absolute value of is defined as |x| = \left\{ \begin{array}{rl} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0. \end{array}\right. :The study of integers is called . ::A prime number is a number that can only be divided by itself and one. If a, b, c, and d are primes then the of abc and c2d is abc2d. (See ) (See ) is defined as repeated addition. 8+8+8 = 3 \cdot 8 . The inverse of multiplication is . But division leads to equations like 3/2=x for which there is no answer among integers. The solution is to generalize to the set of which include fractions (See ). Any number which isnt rational is . : . :(Addition and multiplication are fast but division is slow .) (See ) is defined as repeated multiplication. 7 \cdot 7 \cdot 7 = 7^3 . The inverses of exponentiation are and . :0^0 = 1. See . When a quantity, like the charge of a single electron, becomes so small that it is insignificant we, quite justifiably, treat it as though it were zero. A quantity that can be treated as though it were zero, even though it very definitely is not, is called infinitesimal. If q is a finite ( q \cdot 1 ) amount of charge then dq would be an infinitesimal ( q \cdot 1/\infty ) amount of charge. See Likewise when a quantity becomes so large that a regular finite quantity becomes insignificant then we call it infinite. We would say that the mass of the ocean is infinite ( M \cdot \infty ) . But compared to the mass of the Milky Way galaxy our ocean is insignificant. So we would say the mass of the Galaxy is doubly infinite ( M \cdot \infty^2 ) . Infinity and the infinitesimal are called . Hyperreals behave, in every way, exactly like real numbers. For example, 2 \cdot \infty is exactly twice as big as \infty. In reality, the mass of the ocean is a real number so it is hardly surprising that it behaves like one. Back to top Vectors The one dimensional number line can be generalized to a multidimensional thereby creating multidimensional math (i.e. ). : \mathbb{R}^3 is the \mathbb{R} \times \mathbb{R} \times \mathbb{R}. A is a with and (multiplication of a vector and a belonging to a . :If {\mathbf e_1} , {\mathbf e_2} , {\mathbf e_3} are :and {\mathbf u} , {\mathbf v} , {\mathbf x} are arbitrary vectors then we can (and usually do) write: ::'' \mathbf{u} = u_1 \mathbf{e_1} + u_2 \mathbf{e_2} + u_3 \mathbf{e_3} = \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix} '' ::'' \mathbf{v} = v_1 \mathbf{e_1} + v_2 \mathbf{e_2} + v_3 \mathbf{e_3} = \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix} '' ::'' \mathbf{x} = x_1 \mathbf{e_1} + x_2 \mathbf{e_2} + x_3 \mathbf{e_3} = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} '' The norm of vector \mathbf{v} is denoted \|\mathbf{v}\|. The double bars are used to avoid confusion with the absolute value of the function. :In the norm doesnt depend on the choice of coordinate system. As a result, rigid objects can rotate in Euclidean space. See proof of the to the right. :: \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} Back to top Dot product (a ): \mathbf{u} \cdot \mathbf{v} = \| \mathbf{u} \|\ \| \mathbf{v}\| \cos(\theta) = u_1 v_1 + u_2 v_2 + u_3 v_3 : \mathbf{u}\cdot\mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} = \begin{bmatrix}u_1 v_1 + u_2 v_2 + u_3 v_3 \end{bmatrix} :Only parallel components multiply. :::Two vectors are orthogonal if \mathbf{u \cdot v} = 0. ::::In Euclidean space \|\mathbf{v}\|^2 = \mathbf{v}\cdot\mathbf{v}. Back to top Cross product : \mathbf a\times\mathbf b=\|\mathbf a\|\|\mathbf b\|\sin(\theta)\,\mathbf n Unlike the dot product, it is only defined in \R^3 (that is, three dimensions). It is commonly used in physics, engineering, vector calculus, and linear algebra. Back to top Functions " that takes an input x'', and returns a single corresponding output ''f(x'').]] ''f in the Cartesian plane, consisting of all points with coordinates of the form . The property of having one output for each input is represented geometrically by the fact that each vertical line (such as the yellow line through the origin) has exactly one crossing point with the curve.]] From Wikipedia:Function (mathematics) In mathematics, a '''function'The words map or mapping, transformation, correspondence, and operator are often used synonymously. . is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function f(x)=x^2 that relates each real number x'' to its square ''x''2. The output of a function ''f corresponding to an input x'' is denoted by ''f(x'') (read "''f of x''"). In this example, if the input is −3, then the output is 9, and we may write . See Tutorial:Evaluate by Substitution. Likewise, if the input is 3, then the output is also 9, and we may write . (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function. Back to top Euclids "common notions" : From Wikipedia:Euclidean geometry: Things that do not differ from one another are equal to one another Things that are equal to the same thing are also equal to one another | then a=c |} If equals are added to equals, then the wholes are equal | then a+c=b+d |} If equals are subtracted from equals, then the remainders are equal | then a-c=b-d |} The whole is greater than the part. Back to top Elementary algebra : From Wikipedia:Elementary algebra: Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers. Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations ( , , , and ). For example, *Added terms are simplified using coefficients. For example, x + x + x can be simplified as 3x (where 3 is a numerical coefficient). *Multiplied terms are simplified using exponents. For example, x \times x \times x is represented as x^3 *Like terms are added together,Andrew Marx, Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores, Publisher Kaplan Publishing, 2007, , 9781419552885, 288 pages, page 51 for example, 2x^2 + 3ab - x^2 + ab is written as x^2 + 4ab , because the terms containing x^2 are added together, and, the terms containing ab are added together. *Brackets can be "multiplied out", using . For example, x (2x + 3) can be written as (x \times 2x) + (x \times 3) which can be written as 2x^2 + 3x *Expressions can be factored. For example, 6x^5 + 3x^2 , by dividing both terms by 3x^2 can be written as 3x^2 (2x^3 + 1) For any function f , if a=b then the following four rules apply: # f(a) = f(b) # a + c = b + c # ac = bc # a^c = b^c A typical algebra problem would be to solve for x: :: 5x - 3x - 7 = 4y + 3 :Using rule number 2: :: 5x - 3x - 7 + 7 = 4y + 3 + 7 :we get :: 5x - 3x = 4y + 10 :Factoring out x :: (5 - 3)x = 4y + 10 :we get :: 2x = 4y + 10 :Using rule number 3 :: \frac{2x}{2} = \frac{4y + 10}{2} :we get :: x = 2y + 5 Back to top Trigonometry A right triangle is a triangle with gamma=90 degrees. The Pythagorean theorem posits that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of both legs. : c^2=a^2+b^2 , where c is the length of the side opposite the right angle (the hypotenuse). For small values of x, sin x ≈ x. (If x is in radians). Back to top Polynomials :From Wikipedia:Polynomial: A can always be written in the form : polynomial = Z(x) = a_0 + a_1 x + a_2 x^2 + \dotsb + a_{n-1}x^{n-1} + a_n x^n where a_0, \ldots, a_n are constants called coefficients and n'' is the of the polynomial. :A is a polynomial of degree one. Each individual is the product of the and a variable raised to a nonnegative integer power. :A has only one term. :A has 2 terms. A root (or zero) of a function is a value of x for which Z(x)=0. :: Z(x) = a_n(x - z_1)(x - z_2)\dotsb(x - z_n) :: :The roots of the formula ax^2+bx+c=0 are given by the : :: (x+y)^n = {n \choose 0}x^n y^0 + {n \choose 1}x^{n-1}y^1 + {n \choose 2}x^{n-2}y^2 + \cdots + {n \choose n-1}x^1 y^{n-1} + {n \choose n}x^0 y^n, :Where \binom{n}{k} = \frac{n!}{k! (n-k)!}. See x^2 - y^2 = (x + y)(x - y) The states that the remainder of the division of a polynomial Z(x) by the linear polynomial x-a is equal to Z(a). See . Determining the value at Z(a) is sometimes easier if we use ( ) by writing the polynomial in the form : Z(x) = a_0 + x(a_1 + x(a_2 + \cdots + x(a_{n-1} + x(a_n)))). Back to top Elementary calculus Integration :See also: and The ' ' is a generalization of multiplication. :For example: a unit mass dropped from point x2 to point x1 will release energy. (A unit mass is a mass of one unit). :The usual equation is a simple multiplication. We just multiply the mass times the strength of gravity times the distance that the object falls and the result is how much energy is released: :: 1 \cdot gravity \cdot (x_2 - x_1) = energy :But that equation cant be used if the strength of gravity is itself a function of x. :The strength of gravity at x1 would be different than it is at x2. :And in reality gravity really does depend on x (x is the distance from the center of the earth): :: gravity(x) = 1/x^2 (See .) :However, the corresponding is easily solved: :: \int_{x_1}^{x_2} gravity(x) \cdot dx The fundamental theorem of Calculus is: :: \int_{x_1}^{x_2} f(x) \cdot dx \quad = \quad F(x_2)-F(x_1) :where F(x) is the . ( ) :: \int f(x) \cdot dx = F(x) Finding the indefinite integral is easy: :: \int k \cdot x^y \cdot dx \quad = \quad k \cdot \int x^y \cdot dx \quad = \quad k \cdot \frac{x^{y+1}}{y+1} :where ''k and y'' are arbitrary constants. (Units (feet, mm...) behave exactly like constants.) :and most conveniently: :: \int \bigg (f(x) + g(x) \bigg) \cdot dx = \int f(x) \cdot dx + \int g(x) \cdot dx The integral of a function is equal to the area under the curve. :When the "curve" is a constant (in other words, k•x0) then the integral reduces to ordinary multiplication. Back to top Finite difference From Wikipedia:Finite difference: The slope of a function at point x is approximately: :: \frac{\Delta y}{\Delta x} = \frac{f(x+\Delta x)-f(x)}{\Delta x} :where ::Δx is a small change in the value of x. ::Δy is the corresponding change in y. The smaller Δx becomes the more accurate the approximation becomes. When Δx becomes so small that it is infinitesimal then we denote it dx. Back to top Differentiation Differentiation is the opposite of integration just as division is the opposite of multiplication. ::The of the integral of f(x) is just f(x). The derivative of a function at any point is equal to the slope of the function at that point. :: f'(x)=\frac{dy}{dx} = \frac{f(x+dx)-f(x)}{dx}. The equation of the line tangent to a function at point a is :: y(x) = f(a) + f'(a)(x-a) The derivative of f(x) where f(x) = k•xy is :: f'(x) = {df \over dx} = {d(k \cdot x^y) \over dx} \quad = \quad k \cdot {d(x^y) \over dx} \quad = \quad k \cdot y \cdot x^{y-1} :However there is one exception that you do need to know about. ::The derivative of k \cdot x^0 is k \cdot 0 \cdot x^{-1} = 0 :If the derivative of x0 is not x-1 then what is the integral of x-1? ::The integral of x^{-1} is ln(x) ex = y = dy/dx dx = dy/y = 1/y * dy ∫ (1/y)dy = ∫ dx = x = ln(y) . See And most conveniently: :: (f(x) + g(x))' = f'(x) + g'(x) Examples: :: \frac{d(e^x)}{dx} = e^x :: \frac{d(sin(x))}{dx} = cos(x) :: \frac{d(cos(x))}{dx} = -sin(x) Back to top Taylor & Maclaurin series If we know the value of a at x=0 (smooth means all its derivatives are ) and we also know the value of all of its derivatives at x=0 then we can determine the value at any other point ''x by using the . ("!" means ) : a_0 x^0 + a_1 x^1 + a_2 x^2 + a_3 x^3 \cdots \quad \text{where} \quad a_n = {f^{(n)}(0) \over n!} The proof of this is actually quite simple. Plugging in a value of x=0 causes all terms but the first to become zero. So, assuming that such a function exists, a0 must be the value of the function at x=0. Simply differentiate both sides of the equation and repeat for the next term. And so on. We can easily determine the Maclaurin series expansion of the e^x (because it is equal to its own derivative). : e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = {x^0 \over 0!} + {x^1 \over 1!} + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots And and (because cosine is the derivative of sine which is the derivative of -cosine) : \cos x = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots : \sin x = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots Back to top Fourier Series The Maclaurin series cant be used for a discontinuous function like a square wave because it is not differentiable. But remarkably we can use the to expand it or any other into an infinite sum of sine waves each of which is fully ! Back to top Partial derivatives and generalize derivatives and integrals to multiple dimensions. The partial derivative with respect to one variable \frac{\part f(x,y)}{\part x} is found by simply treating all other variables as though they were constants. Multiple integrals are found the same way. Gradient Numbers are called scalars to distinguish them from vectors. A scalar function f(x) outputs a scalar number for each input value of x. Let f(x, y, z) be a 3 dimensional . :(An elevation map would be an example of a 2 dimensional scalar function because it assigns a scalar number (the height) to each point on the 2 dimensional map.) The of a scalar function is a vector that points "downhill" with a magnitude equal to the of the function at that point. The gradient of a scalar function always goes downhill and therefore never goes in circles. :: \operatorname{grad}(f) = \nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} = \mathbf{F} The function is a scalar function. But \mathbf{F} is not a scalar function. \mathbf{F} is a vector field. That is why it is written in bold text. A vector field for the movement of air on Earth would associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the wind speed.Wikipedia:Vector field Divergence The of the vector field is positive at places where flux lines begin (at the tops of hills) and negative at places where flux lines end (at the bottoms of holes) and zero everywhere else. : \operatorname{div}\,\mathbf{F} = {\color{red} \nabla\cdot\mathbf{F} } = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot (F_x,F_y,F_z) = \frac{\partial F_x}{\partial x} +\frac{\partial F_y}{\partial y} +\frac{\partial F_z}{\partial z}. An even better way to represent vector fields is using . Flux lines are unbroken lines that extend from points of positive divergence all the way to the points of negative divergence (or to infinity if there are no points of negative divergence). The intensity (for example wind speed) is indicated by the density of the flux lines. The more the lines are crowded together the greater the intensity (wind speed). Flux lines have a tendency to repel one another. Electric field lines begin at and end at . Curl The of a vector field describes how much the flux lines are twisted. The curl of the gradient of a scalar function is always zero but that is not true for the curl of any arbitrary vector field. The flux lines of an arbitrary vector field may even go in circles. The flux lines of a magnetic field always go in circles. : \text{curl} (\mathbf{F}) = {\color{blue} \nabla \times \mathbf{F} } = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ F_x & F_y & F_z \end{vmatrix} : \text{curl}( \mathbf{F}) = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \mathbf{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{k} Green's theorem You can think of each field line as ending in a single unit of charge. states that if you want to know how many field lines exit a region then you can either count how many lines cross the boundary (perform a line integral) or you can simply count the number of charges within that region. See . A version of Green's theorem also works for curl. Green's theorem is an extremely important result that is widely used in more advanced mathematics. Green's theorem might seem like a trivial result that is so obvious that it isnt even worth stating but in more advanced mathematics it is used in places and in ways that are far from obvious. Back to top Elementary physics :Highly recommend: Thinking Physics Is Gedanken Physics by Lewis Carroll Epstein Physics is all calculus. Trying to do physics without calculus is like trying to run a race with one leg. If the position of an object as a function of time is given by :: position = p(t) then the velocity is given by :: v = \frac{dp}{dt} and the acceleration is given by :: a = \frac{dv}{dt} = \frac{d^2p}{dt^2} Velocity has units of distance per time. Acceleration has units of distance per time per time. The momentum of an object is equal to the mass times the velocity :: Momentum = m v The kinetic energy of an object is equal to half the mass times the velocity squared :: E = \frac{m v^2}{2} In the image below an object of mass 1 moving with velocity 4 strikes a stationary object with equal mass. The total horizontal momentum, total vertical momentum, and total energy are unchanged. In other words, energy and momentum are . A force of 1 Newton acting on a mass of 1 kg will cause that mass to accelerate at 1 m/s2 :: F = m a :After the object has moved a distance of 2 meters it will have a kinetic energy of :: E = F \cdot distance = 1 \, Newton \cdot 2 \, meters = 2 \, Joules Gravity on Earth causes all free falling objects to accelerate downward at 9.8 m/s2 Back to top Intermediate mathematics :See Intermediate mathematics References